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            "13": {
                "pageid": 13,
                "ns": 0,
                "title": "Resolution",
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                        "*": "The ''pixel resolution'' of a bitmap image is the number of pixels in both dimensions: width and height. It is typically expressed as, for instance, 640x480: this means a width of 640 pixels and a height of 480 pixels.\n\nThis is not to be mistaken with ''spatial resolution'' a.k.a. ''definition'' a.k.a. ''dot density'' which measures how close are the pixels in a realization of a bitmap, either in print or on a screen. Typically a laser printer's definition is at least 300 dots per inch.\n\nIn this wiki, ''resolution'' refers to pixel resolution.\n\nSee [http://en.wikipedia.org/wiki/Image_resolution| the corresponding article] on Wikipedia (December 2014) for more information about these notions of resolution and other variants."
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            "11": {
                "pageid": 11,
                "ns": 0,
                "title": "Tc",
                "revisions": [
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                        "*": "A tc is a data structure encoding a complex number and a variation of this number. The operations I define on these numbers are the same as most operations on complex numbers.\nThe corresponding C++ library of functions that I designed really made writing my programs easier, especially for [[distance estimator|distance estimator methods]].\n\n== Overview ==\n\nA tc is a complex number z together with a complex vector dz attached.\nThe notation dz is probably not the best but it is to mimic physicist notation, like\n\\[(z+dz)\\times(w+dw)=w\\,z+(w\\,dz+z\\,dw)+\\text{neglected}\\]\nor\n\\[\\cos(z+dz)=\\cos(z)-\\sin(z)dz.\\]\nA mathematical reinterpretation in terms of ''jets'' is given below.\n\n== Operator overloading ==\n\nC++ allows [https://en.wikipedia.org/wiki/Operator_overloading operator overloading]. In other words, you can use instructions like d=c+a*b in your programs, with any kind type of objects for a,b,c,d.\n\nThis is why it is much easier to program with complex numbers in C++ than in many other languages. Compare C++\n<syntaxhighlight lang=\"cpp\">\nz=u*z*(z*z+cos(c*z)+d);\n</syntaxhighlight>\nwith Java (no operator overloading)\n<syntaxhighlight lang=\"java\">\nz=mul(u,mul(z,add(mul(z,z),add(cos(mul(c,z)),d))));\n</syntaxhighlight>\nor with C\n<syntaxhighlight lang=\"c\">\nmul(w,c,z);\ncos(w,w);\nadd(w,w,d);\nmul(v,z,z);\nadd(w,w,v);\nmul(w,z,w);\nmul(z,u,w);\n</syntaxhighlight>\n\n== Operations ==\n\nOne defines operations on tc objects as follows:\n\\[(z,dz) + (w,dw) = (z+w,dz+dw)\\]\n\\[(z,dz) \\times (w,dw) = (z\\,w,w\\,dz+z\\,dw)\\]\n\\[-(z,dz) = (-z,-dz)\\]\n\\[1/(z,dz) = (1/z,-dz/z\u02c62)\\]\n\\[\\overline{(z,dz)} = (\\overline{z},\\overline{dz})\\]\n\\[\\exp(z,dz)=(\\exp(z),\\exp(z)dz)\\]\n\\[\\cos(z,dz)=(\\cos(z),-\\sin(z)dz)\\]\n\\[etc...\\]\n\n== Tangent space and jets ==\n\nOne can imagine that a tc pair (z,dz) represents a moving point, starting from z and moving at speed dz:\n$z(t)=z+dz\\,t+o(t)$ .\nIn other words it is an element in the tangent space TC of the complex numbers field C.\nThis is why I chose the name tc for the C++ class.\nIt can also be considered as 1-jets (can be generalized to higher degree power series expansions, like the $(z,b,c)$ being a 2-jet representing $z(t)=z+b\\,t+c\\,t^2+o(t^2)$).\nJets are differential geometry object, i.e. there are specific formulae for computing how their expression (coordinates) changes when changing variables.\n\n== Derivatives ==\n\nThe tc objects compute derivatives for you!\n\n(Let us be clear: it does not give you a formal formula. I meant it gives you a numeric value of f' at a given numeric value of the variable.)\n\nSay you defined Z=(z,1), computed W=cos(Z\u00d7Z) and got W=(a,b). Then a=cos(z\u00d7z) and b=the derivative \u2202a/\u2202z at z: you did not need to determine that $\\partial \\cos(z^2)/\\partial z=-2z\\sin(z^2)$, the class computed b iteratively for you.\n\nMaybe it is more convincing with a more complicated example: you need the derivative at a given value of z of the quantity $\\frac{\\displaystyle \\log z+\\cos\\frac{1+e^\\sqrt z}{1-\\sin(z)(1+z)}}{\\displaystyle \\sqrt{z^3-2z+12}}$? Here is a recipe: define the tc object Z=(z,1), compute the quantity above using Z, and take the dz part of the result.\n\n== Implementation ==\n\nThe code below is a possible implementation. \nOnly the functions I most use are implemented (not cosh, not ==, etc... for instance).\nIt not optimized and there is no specific handling of division by 0 and overflow, which I imagine may be problematic.\n\n<syntaxhighlight>\n#include <complex>\n\ntypedef double reel;\ntypedef std::complex<reel> cplx;\n\nclass tc {\n public : // all the members below are accessible by users of the package\n  // data members\n  cplx z;\n  cplx dz;\n  // member functions (declarations only, the code is given below*)\n  // *: for this C++ unses the following (ugly) syntax \n  //    return-type class-name::function-name(parameters) { code; }\n  // constructors\n  tc(cplx=cplx(0,0), cplx=cplx(0,0)); // default constructor\n  tc(const tc&); // copy constructor\n  // assigment operations : unlike math operations and math functions, they /have/ to belong to the class\n  tc& operator=(const tc&); \n  const reel& operator=(const reel&);\n  const cplx& operator=(const cplx&);\n};\n\n// now we give the code of the member functions\n\n// a constructor\ntc::tc(cplx point, cplx vecteur)\n{\n z = point;\n dz = vecteur;\n}\n\n// the copy constructor\ntc::tc(const tc& t)\n{\n z = t.z;\n dz = t.dz;\n}\n \n// assignment operators\n\ntc& tc::operator=(const tc& t)\n{\n z = t.z;\n dz = t.dz;\n return *this;\n}\n\nconst reel& tc::operator=(const reel& r)\n{\n z = r;\n dz = 0;\n return r;\n}\n\nconst cplx& tc::operator=(const cplx& c)\n{\n z = c;\n dz = 0;\n return c;\n}\n\n// non-member functions : usual math operators and math functions\n\ntc operator*(const tc& a, const tc& b) {\n return tc(a.z*b.z, a.z*b.dz+a.dz*b.z);\n}\n\ntc operator*(const reel& a, const tc& b) {\n return tc(a*b.z, a*b.dz);\n}\n\ntc operator*(const tc& a, const reel& b) {\n return tc(a.z*b, a.dz*b);\n}\n\ntc operator+(const tc& a, const tc& b) {\n return tc(a.z+b.z,a.dz+b.dz);\n}\n\ntc operator+(const reel& a, const tc& b) {\n return tc(a+b.z,b.dz);\n}\n\ntc operator+(const tc& a, const reel& b) {\n return tc(a.z+b,a.dz);\n}\n\ntc operator-(const tc& a) {\n return tc(-a.z,-a.dz);\n}\n\ntc operator-(const tc& a, const tc& b) {\n return tc(a.z-b.z,a.dz-b.dz);\n}\n\ntc operator-(const reel& a, const tc& b) {\n return tc(a-b.z,-b.dz);\n}\n\ntc operator-(const tc& a, const reel& b) {\n return tc(a.z-b,a.dz);\n}\n\ntc operator/(const tc& a, const tc& b) {\n return tc(a.z/b.z,a.dz/b.z-a.z*b.dz/(b.z*b.z));\n}\n\ntc operator/(const reel& a, const tc& b) {\n return tc(a/b.z,-a*b.dz/(b.z*b.z));\n}\n\ntc operator/(const tc& a, const reel& b) {\n return tc(a.z/b,a.dz/b);\n}\n\ntc exp(const tc& a) {\n cplx aux=exp(a.z);\n return tc(aux,aux*a.dz);\n}\n\ntc sin(const tc& a) {\n return((exp(a*tc(0,1))-exp(a*tc(0,-1)))*tc(0,-0.5));\n}\n\ntc cos(const tc& a) {\n return((exp(a*tc(0,1))+exp(a*tc(0,-1)))*0.5);\n}\n\ntc tan(const tc& a) {\n  return(sin(a)/cos(a));\n}\n\ntc log(const tc& a) {\n return tc(log(a.z),a.dz/a.z);\n}\n\ntc sqrt(const tc& a) {\n return exp(0.5*log(a));\n}\n\ntc conj(const tc& a) { // CAUTION : here when interpreting the dz part, only reel values of 'time' make sense\n return tc(conj(a.z), conj(a.dz));\n}\n\nstatic const cplx I(0,1);\n</syntaxhighlight>"
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